Would you explain the difference in the Old Covenant and the New Covenant? I have been reading in Jeremiah 31:31-34 [31] Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: [32] Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: [33] But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. [34] And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
See All... and need a clearer understanding of all of it.

For easy reference, here is the text of the passage:

Jeremiah 31:31-34 [31] Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: [32] Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: [33] But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. [34] And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
See All... Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.

The Old Covenant is the law of Moses as given to the children of Israel. In Exodus 24:1-8 [1] And he said unto Moses, Come up unto the LORD, thou, and Aaron, Nadab, and Abihu, and seventy of the elders of Israel; and worship ye afar off. [2] And Moses alone shall come near the LORD: but they shall not come nigh; neither shall the people go up with him. [3] And Moses came and told the people all the words of the LORD, and all the judgments: and all the people answered with one voice, and said, All the words which the LORD hath said will we do. [4] And Moses wrote all the words of the LORD, and rose up early in the morning, and builded an altar under the hill, and twelve pillars, according to the twelve tribes of Israel. [5] And he sent young men of the children of Israel, which offered burnt offerings, and sacrificed peace offerings of oxen unto the LORD. [6] And Moses took half of the blood, and put it in basons; and half of the blood he sprinkled on the altar. [7] And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient. [8] And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the LORD hath made with you concerning all these words.
See All..., we have the record of the formal making of this covenant between the Lord and the Israelites. Moses sprinkled the blood of the covenant on the altar and on the people and the people promised to keep the words of the book of the covenant. This is the covenant mentioned in Jeremiah 31:32Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD:
See All... (see above).

The New Covenant is promised in Jeremiah 31:31-34 [31] Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: [32] Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: [33] But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. [34] And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
See All... and also in Ezekiel 37:21-28 [21] And say unto them, Thus saith the Lord GOD; Behold, I will take the children of Israel from among the heathen, whither they be gone, and will gather them on every side, and bring them into their own land: [22] And I will make them one nation in the land upon the mountains of Israel; and one king shall be king to them all: and they shall be no more two nations, neither shall they be divided into two kingdoms any more at all. [23] Neither shall they defile themselves any more with their idols, nor with their detestable things, nor with any of their transgressions: but I will save them out of all their dwellingplaces, wherein they have sinned, and will cleanse them: so shall they be my people, and I will be their God. [24] And David my servant shall be king over them; and they all shall have one shepherd: they shall also walk in my judgments, and observe my statutes, and do them. [25] And they shall dwell in the land that I have given unto Jacob my servant, wherein your fathers have dwelt; and they shall dwell therein, even they, and their children, and their children's children for ever: and my servant David shall be their prince for ever. [26] Moreover I will make a covenant of peace with them; it shall be an everlasting covenant with them: and I will place them, and multiply them, and will set my sanctuary in the midst of them for evermore. [27] My tabernacle also shall be with them: yea, I will be their God, and they shall be my people. [28] And the heathen shall know that I the LORD do sanctify Israel, when my sanctuary shall be in the midst of them for evermore.
See All.... The new covenant is also verified in Hebrews 8:8-13 [8] For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah: [9] Not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they continued not in my covenant, and I regarded them not, saith the Lord. [10] For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people: [11] And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest. [12] For I will be merciful to their unrighteousness, and their sins and their iniquities will I remember no more. [13] In that he saith, A new covenant, he hath made the first old. Now that which decayeth and waxeth old is ready to vanish away.
See All.... However, the Jeremiah reference is the key passage. I will use it to make several important points about the New Covenant.

Historically and primarily, the New Covenant is promised to the Jewish people. Jeremiah 31:31Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah:
See All... is very specific. It will be made with the house of Israel (the northern kingdom) and with the house of Judah (the southern kingdom). There is no way to spiritualize this promise and keep the meaning intact. The Lord is referring to the Jewish people. They have not yet received the New Covenant. This is still in the future for them. This is the covenant Paul has in mind in the following passage where he is clearly referring to the earthly Israelites.

Romans 11:25-29 [25] For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. [26] And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: [27] For this is my covenant unto them, when I shall take away their sins. [28] As concerning the gospel, they are enemies for your sakes: but as touching the election, they are beloved for the father's sakes. [29] For the gifts and calling of God are without repentance.
See All... For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: For this is my covenant unto them, when I shall take away their sins. As concerning the gospel, they are enemies for your sakes: but as touching the election, they are beloved for the fathers' sakes. For the gifts and calling of God are without repentance.

However, I will add one addendum. I believe that the Gentiles of this age have been enabled to partake of some of the privileges of the New Covenant through the cross of Christ. Specifically, this is distinguished in the English Bible as the New Testament. So, we partake of some of the blessings that will later come on the Israelites through the blood of Christ. This is possible because the blood of Christ on the cross is the blood of the covenant (Hebrews 13:20Now the God of peace, that brought again from the dead our Lord Jesus, that great shepherd of the sheep, through the blood of the everlasting covenant,
See All...); that is, it is the basis of the future relationship of the Israelites in the New Covenant. The New Covenant with the Jews is based on the blood of Christ. But this in no way replaces the promise of the New Covenant to the Jewish people as taught in Romans 11:25-27 [25] For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. [26] And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: [27] For this is my covenant unto them, when I shall take away their sins.
See All... and other places.

The New Covenant has several provisions:

• God's law will be written in their hearts (Jeremiah 31:33But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
  See All...). Ezekiel 37:26-27 [26] Moreover I will make a covenant of peace with them; it shall be an everlasting covenant with them: and I will place them, and multiply them, and will set my sanctuary in the midst of them for evermore. [27] My tabernacle also shall be with them: yea, I will be their God, and they shall be my people.
  See All... refers to this "new heart" and "new spirit" that will cause them to walk in God's statutes and keep His judgments.
• Their relationship with God will be perfect. He will be their God and they will be His people (Jeremiah 31:33But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
  See All...). This is a common expectation in the Old Testament (Exodus 6:7And I will take you to me for a people, and I will be to you a God: and ye shall know that I am the LORD your God, which bringeth you out from under the burdens of the Egyptians.
  See All...; Leviticus 26:12And I will walk among you, and will be your God, and ye shall be my people.
  See All...; Jeremiah 24:7And I will give them an heart to know me, that I am the LORD: and they shall be my people, and I will be their God: for they shall return unto me with their whole heart.
  See All...; 30:22
  See All...; 31:1
  See All...; 32:38
  See All...; Ezekiel 11:20That they may walk in my statutes, and keep mine ordinances, and do them: and they shall be my people, and I will be their God.
  See All...; 37:27
  See All...; Zechariah 13:9And I will bring the third part through the fire, and will refine them as silver is refined, and will try them as gold is tried: they shall call on my name, and I will hear them: I will say, It is my people: and they shall say, The LORD is my God.
  See All...) and is finally fulfilled in the New Jerusalem (Revelation 21:3And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God.
  See All...) when all the righteous will be included.
• Their knowledge of God will be universal (Jeremiah 31:34And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
  See All...). They will all know Him, from the least to the greatest. By the way, this provision does not apply to the age of grace since all do not know Him today and this demonstrates the fact that the New Covenant will not be completely fulfilled until it is fulfilled in the Jewish people--the people to whom it was originally promised.
• Their iniquities and sins will be completely forgiven (Jeremiah 31:34And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
  See All...). Romans 11:25-27 [25] For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. [26] And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: [27] For this is my covenant unto them, when I shall take away their sins.
  See All... shows how this promise directly applies to Israel.

The Old Covenant of the law failed because of the failure of the Jews to obey it. However, God will overcome this failure by the future application of the New Covenant when He brings the Jews into perfect relationship with Himself. Yet, even today through the blood of the Lord Jesus Christ and the New Testament, we begin to understand the greatness and blessings of the New Covenant. It is only important that we do not think that we have fulfilled the New Covenant in any complete way. That remains to be fulfilled in the house of Israel and the house of Judah.